Abstract

We present a multivariable setting for Lax-Phillips scattering and for conser-

vative, discrete-time, linear systems. The evolution operator for the Lax-Phillips

scattering system is an isometric representation of the Cuntz algebra, while the

nonnegative time axis for the conservative, linear system is the free semigroup on d

letters. The correspondence between scattering and system theory and the roles of

the scattering function for the scattering system and the transfer function for the

linear system are highlighted. Another issue addressed is the extension of a given

representation of the Cuntz-Toeplitz algebra (i.e., a row isometry) to a representa-

tion of the Cuntz algebra (i.e., a row unitary); the solution to this problem relies on

an extension of the Szego factorization theorem for positive Toeplitz operators to

the Cuntz-Toeplitz algebra setting. As an application, we obtain a complete set of

unitary invariants (the characteristic function together with a choice of "Haplitz"

extension of the characteristic function defect) for a row-contract ion on a Hilbert

space.

Received by the editor November 13, 2002 and in revised form September 28, 2004.

2000 Mathematics Subject Classification. Primary: 47A48; Secondary: 13F25, 47A40, 47L30,

47L55, 93C05.

Key words and phrases, formal power series, noncommuting indeterminants, unitary colliga-

tion, incoming and outgoing space, scattering function, row isometry, row contraction, functional

model.

The first author is supported by NSF grant DMS-998763; both authors are supported by a

grant from the US-Israel Binational Science Foundation.